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Practice Questions
Test your understanding with targeted questions related to the topic.
Question 1
Easy
What is the condition for a matrix to be diagonalizable?
💡 Hint: Consider the relationship between eigenvalues and the independence of eigenvectors.
Question 2
Easy
Define eigenvalue in your own words.
💡 Hint: Think about what happens to vectors when transformed by matrices.
Practice 4 more questions and get performance evaluation
Interactive Quizzes
Engage in quick quizzes to reinforce what you've learned and check your comprehension.
Question 1
What does it mean for a matrix to be diagonalizable?
- It can be reduced to a zero matrix
- It has a complete set of eigenvectors
- It is always symmetric
💡 Hint: Think of the properties of the eigenvectors.
Question 2
True or False: All matrices with repeated eigenvalues are non-diagonalizable.
- True
- False
💡 Hint: Consider examples of matrices with distinct eigenvalues.
Solve and get performance evaluation
Challenge Problems
Push your limits with challenges.
Question 1
Given the matrix A = [2 3; 2 4], demonstrate if A is diagonalizable and find its eigenvalues and eigenvectors.
💡 Hint: Focus on the linearly independent nature of the eigenvectors.
Question 2
Prove the Jordan form is useful for non-diagonalizable matrices by providing an example and explaining the process.
💡 Hint: Think about the roles of blocks in Jordan form.
Challenge and get performance evaluation